As an Australian, I would say a good test of a university's reputation in the US is who it associates with. What I mean is, U.S. institutions don't just let anyone into their exchange programs.
So, who are the good Australian universities partnered with? (notes, or courses where the exchange applies, in brackets)
ANU: University of Pennsylvania (Ivy League), UC system, UNC system, Texas at Austin, also Oxford University (Australia's only partnership here)
Melbourne: Boston College, Carnegie Mellon (for eng, science), Cornell (Ivy), Duke (Law), Georgetown, NYU (Commerce), UC system, Michigan, UNC, Penn (Ivy), USC (Commerce), Texas at Austin, UVA, Washington U @ St. Louis (Commerce)
Sydney: Boston U, Cornell (Ivy), Duke (Law and business), NYU (Law), Purdue, George Washington, UC system, Penn (Ivy), Texas at Austin
Monash: Boston College, Purdue, UC system
That's just a sample. I would say that a degree from any of the
Group of Eight will be respected in the US. These names aren't going to jump off the page for an adcom, but as with anything in the US, these schools won't hold a talented applicant back.
I would also point out a less well known option, but a very good one for international students.
Bond University is Australia's only private university, and has about 3000 students with up to 50% international when study abroad numbers are at their peak. The university has been highly rated, with Australia's best graduate salaries, and even in the few decades of operation has a Rhodes Scholar and Ivy League graduate admits. If we use the international partnerships test, then it has connections to Duke and Northwestern in the US, both top 10 schools.